development and validation of inverse flight dynamics simulation for helicopter...
TRANSCRIPT
Development and Validation of Inverse Flight Dynamics Simulation for HelicopterManeuver
Raghavendra PrasadGraduate Student
AbhishekAssistant [email protected]
Department of Aerospace EngineeringIndian Institute of Technology Kanpur
Kanpur, India
ABSTRACTThis paper discusses the development, implementation and validation of a robust inverse flight dynamics simulation forhelicopters undergoing steady and unsteady maneuvers. An algorithm based on integration based inverse simulationapproach is implemented which regulates the required vehicle accelerations necessary to track the desired trajectoryby adjusting the controls required by the helicopter. Helicopter dynamics model consists of individual rigid bladeswith flapping dynamics and quasi-steady aerodynamics. The analysis has the option of using airfoilCl andCd lookuptables for non-linear aerodynamic modelling, if such tables are available. The present inverse flight dynamics analysisis verified and validated using available flight test and simulated results for literature for UH-60A Black Hawk andWestland Lynx helicopters respectively. The derived procedure is used to satisfactorily predict the pilot input controlsand shaft orientation angles that are required to fly severalmaneuvers including pull-up, hurdle-hop, pop-up, bob-upand pirouette.
NOTATIONS
u : Control vectoru,v,w : Tanslational velocity in body framep,q,r : Angular velocity in body framex : System state vectory : Output vectorV f : Resultant velocity (m/s)Vmax : Maximum velocity (m/s)Vtip : Blade tip velocity (m/s)V : Velocity (m/s)t : Time (s)tm : Maneuver time (s)h : Altitude (m)m : Helicopter mass (kg)Xe,Ye,Ze : Helicopter position in gravity axesθ0 : Blade collective pitchθ1c : Blade lateral cyclic pitchθ1s : Blade longitudinal cyclic pitchαs : Longitudinal shaft tiltφs : Lateral shaft tilta : Time rate of change of any variableaω : Angular velocity of main rotor (rad/s)Ω: Angular velocity of helicopter (rad/s)
Presented at the 4th Asian/Australian Rotorcraft Forum, IISc,India, November 16–18, 2015. Copyrightc© 2015 by theAsian/Australian Rotorcraft Forum. All rights reserved.
INTRODUCTION
Study of helicopters, for either the handling qualities or theprediction of helicopter blade loads, requires the knowledgeof control inputs necessary to fly / simulate the desired flight.Trim analysis can be performed to predict the input controlsfor steady flight in which controls are obtained by targetingsteady equilibrium of the vehicle. But in case of unsteadymaneuvers, where states are instantaneous and transient, trimanalysis cannot be performed because states are also to bedetermined simultaneously with controls. Derivation of timehistory of pilot input controls required for flying or simulat-ing an unsteady maneuvers is called inverse simulation. Thisis opposite to usual forward simulation where the input con-trols are known and the output response to prescribed inputis of interest. During inverse simulation, the desired outputresponse, such as, trajectory, is known and the input controlsrequired to achieve the desired output response is calculated.
Several successful attempts have been made to use inverseflight dynamic simulation for helicopter application follow-ing the demonstration of its usefulness in deriving controlsfor fixed wing aircraft (Refs.1,2). Several methods have beenstudied by researchers over the years for inverse flight dynam-ics modelling. All these approaches available in literature canbe broadly classified into three categories: 1) differentiationapproach, 2) integration approach and 3) global optimizationbased methods.
Thomson and Bradley (Refs.3,4) were among the first re-
1
searchers to develop an inverse flight dynamics simulation al-gorithm “HELINV” for rotorcraft application. Their approachwas based on numerical differentiation of states at every timestep to evaluate the time derivatives of the states. This allowedthem to transform the governing differential equations to al-gebraic equations which could be solved to obtain the controlangles. Despite being model dependent and sensitive to timestep size this method highlighted the usefulness of inversesimulation as an effective tool in helicopter flight dynamicsby analyzing several unsteady maneuvers.
Hess et al. (Refs.5, 6) formulated integration inversemethod in which states are modified using repeated numeri-cal integration of the equations of motion during a constrainedtime step by keeping the controls constant. At the end of thetime step the controls are modified using Newton Raphsonto reduce the error between the desired and achieved states.While, it is an order of magnitude slower than the differen-tiation based approach, it is independent of rotor dynamicsmodel and allows for its refinement without any need for al-tering the solution procedure. Lin (Ref.7) reported the nu-merical issues such as constrained oscillations associated withthis approach. De Matteis et al. (Ref.8) replaced the Newton-Raphson step with local optimization procedure, eliminatingconstrained oscillations mentioned earlier in this process. Inthis approach, controls are calculated by minimizing a costfunction which is defined based on required performance in-dices. This is done using either classical finite differenceNewton method or Sequential Quadratic Programming (SQP)algorithm. A “two-timescale” method of integration basedinverse simulation was proposed by Avanzini and de Mat-teis (Ref.9) that reduces the computational effort by decou-pling and solving the rotational and translational dynamicsseparately. But this method adds certain level of inaccuraciesto the solution.
Global optimization method, proposed by Celi (Ref.10),operates on a whole family of possible trajectories (and there-fore, of pilot command time histories) unlike the methodsderived earlier. Based of required performance criteria thebest trajectory and corresponding set of controls are cho-sen to carry out a specific maneuver. However, this re-quires significant amount of function evaluations, resultingin significant computational penalty. A general optimiza-tion method studied by Lee and Kim (Ref.11) used time-domain finite element method to discretize the governingequations for blade response and numerical solution for con-trol angles is obtained through least square optimization.Luand Murray-Smith (Ref.12) developed an inverse simulationmethod based on sensitivity-analysis theory in which a Jaco-bian matrix is calculated by solving sensitivity equation.Thismethodology overcomes numerical as well as inputoutput re-dundancy problem that arise in traditional methodologies.
In addition to gradient based method, global optimizationmay be achieved by using genetic algorithm based approach.This obviates the need for calculation of gradients but the con-vergence is typically slower. A genetic algorithm based ap-proach was used for inverse simulation of pirouette and slalom
manoeuvres by Guglieri and Mariano (Ref.13) to success-fully demonstrate the precise tracking of the prescribed tra-jectories with reasonable control inputs. A Hybrid GeneticAlgorithm is also used for helicopter trim research, whichtries to combine the reliability of the Genetic Algorithm withthe speed of the quasi-Newton iterative approach. The cal-culation in started with Genetic Algorithm and switches toquasi-Newton method when a desired fitness level is achieved(Ref. (Ref.14)). This allows the algorithm to converge in alarger range with a faster convergence velocity.
In this paper, focus is on preliminary development of ro-bust inverse flight dynamics simulation for estimating controlangles for wide range of maneuvers performed by helicopters.The integration based approach is used for the present analysisdue to its ease of implementation and independence from he-licopter rotor dynamics models. The long term goal is to pro-gressively refine the rotor dynamics and aerodynamics mod-els within the analysis and understand their effect on the pre-diction capability of the model. The current analysis is firstvalidated by simulation of aggressive pull-up maneuver forUH-60A Black Hawk helicopter, for which flight test data isavailable in public domain. The control prediction capabil-ity of the present analysis is then verified for other maneuverssuch as pop-up and hurdle-hop maneuvers for Westland Lynxhelicopter with available simulated results from literature. Fi-nally, bob-up and pirouette maneuvers are also simulated.
APPROACH
Integration based inverse flight dynamics simulation approachis chosen for present analysis. In this approach vehicle states,in body reference frame, are systematically integrated over aconstrained time step, transformed into earth fixed referenceframe and controls are modified by reducing the error betweensimulated and required velocity at the end of time step usingNewton-Raphson. The main advantage of this procedure ismodel can be separated from simulation algorithm that makesthe procedure very flexible in studying the effect of variousmodelling refinements. This procedure is explained below.
A dynamic system can be described as a function of statesand controls
x = f (x,u) (1)
by using the updated states from the above equation, instanta-neous system output can be derived from the following equa-tion, provided that time history of system controls is given.
y = g(x) (2)
But in a situation where time history of desired outputis known and sequence of controls that drive the system toachieve the given output are to be determined, system dynam-ics must be inverted. The basis for inverse simulation can beobtained by differentiating the output equation with respect totime until the required controls appear in the resulting equa-tion.
y =dgdx
x (3)
2
y =dgdx
f (x,u) (4)
controls can be calculated by inverting the above equation,which takes the form
u = h(x, y) (5)
thus controls can be written as function of system output andcorresponding states.
For the present analysis, system is represented by Newton-Euler equations, shown below, governing the rigid body dy-namics of helicopter. Though these equations are general forany rigid body, the derivation of external forcesX , Y , Z andmomentsL, M, N, acting along the three body fixed axes ofthe helicopter, are carried out using a helicopter rotor dynam-ics model.
Force equilibrium equations
m(u+ qw− rv)+mgsinθ = X (6)
m(v+ ru− pw)−mgsinφ cosθ = Y (7)
m(w+ pv− qu)−mgcosφ cosθ = Z (8)
Moment equilibrium equations
Ixx p− (Iyy − Izz)qr+ Iyz(r2− q2)− Ixz(pq+ r)
+Ixy(pr− q) = L(9)
Iyyq− (Izz− Ixx)pr+ Ixz(p2− r2)− Ixy(qr+ p)
+Iyz(pq− r) = M(10)
Izzr− (Ixx − Iyy)pq+ Ixy(q2− p2)− Iyz(pr+ q)
+Ixz(qr− p) = N(11)
Kinematics equations
p = φ − ψ sinθ (12)
q = θ cosφ + ψ sinφcosθ (13)
Full six degrees of freedom of the vehicle are consideredfor current analysis: all three translational and rotational de-grees of freedom. But it should be noted that the yaw degreeof freedom does not play a significant role for the maneuversbeing analyzed in this research which primarily involve mo-tion in vertical-longitudinal plane and any out of plane motionis of small magnitude.
In case of a conventional helicopter, the key system statesof interest are
x = u,v,w, p,q,r (14)
and the control inputs that need to be determined are
u = θ0,θ1c,θ1s,θtrT (15)
and the desired outputy would be the maneuver trajectory thatis to be performed by aircraft. The definition and modeling ofvarious maneuvers is discussed later in text.
Integration Inverse Method
The simulation is initiated by dividing the entire trajectory forthe maneuver into small steps. Then at each instance of time,an estimate of the change in the amplitude of control displace-ment required to move the aircraft to the next point is carriedout. It is assumed that the controls are constant for each timeinterval [tk,tk+1]. Initial guesses of the controlsu(tk) are usedfor the forward simulation. The outputy(tk+1) obtained fromthe simulation is compared with the desired outputy∗(tk+1).Based on the errors, the guessed controls are modified using,for example, Newton’s method. The process repeats itself un-til the simulation result converges to the desired trajectory attk+1. Then the analysis is moved to the next time step.
A step by step procedure describing the integration inverseapproach is included below:
1. The set of equations of motion is derived.
2. The initial value of the State Vector is calculated.
3. The Desired Trajectory is defined.
4. The trajectory is descretized into Constrained TimeSteps.
5. Each constrained time step is further descretized into El-emental Time Steps.
6. The initial guess of the control vector is made by solvingthe trim problem for initial steady state.
7. The State vector is updated by forward simulation to ob-tain the actual output at the Constrained Time Step, con-sidering the control vector as constant throughout the en-tire Constrained Time Step.
8. The deviation of the actual trajectory from the desiredtrajectory (at Constrained Time Step) is calculated andthe Control Vector is updated.
9. The above mentioned steps are repeated until the devi-ation of the actual trajectory from the desired trajectoryreaches a predetermined minute value, which results inthe Required Control Vector that is to be applied initially(i.e., at the previous Constrained Step) to achieve approx-imately desired trajectory at the Constrained Step-1 (i.e.,at the new Constrained Step).
10. The above iterative procedure is continued for the re-maining constrained steps. A flowchart describing theimplementation of integration inverse simulation proce-dure to unsteady maneuver is shown in Fig.1.
Where the notations have their usual meaning, e.g.φ , θ ,andψ respectively denote the roll, pitch, and yaw attitudes ofthe helicopter.
3
Fig. 1. Flow chart depicting integration inverse simulationprocedure
Helicopter Dynamics Model
The main rotors of the two helicopters, UH-60A Black Hawkand Westland Lynx, being simulated are modeled as fully ar-ticulated rotors with rigid blades with hinge offset and hav-ing only the flap degree of freedom. Appropriate root springis selected to match the first flap frequency. The individ-ual blades are modeled with linear twist and root cut-out.The rotor longitudinal pre-shaft tilt angle is also considered.Though a standard symmetric airfoil with lift curve slope of5.73 is used over the entire rotor blade for Westland Lynx he-licopter, two different airfoil tables corresponding to SC1095and SC1094R8, are used for UH-60A Black Hawk rotor.
Tail rotor model is a simple model based on Blade ElementTheory (BET) while the tail rotor inflow is calculated usinguniform inflow. The side force contribution of the tail rotorisincluded in the equation of motion to compensate for the mainrotor torque required to maintain stable yaw.
Since, the lookup tables for the airfoils used on the UH-60A Black Hawk Helicopter is available in public domain,it is used for non-linear quasi-steady aerodynamic calcula-tions during this analysis. The detailed rotor geometry andblade properties data for UH-60A is taken from (Refs.15–18).
The flight test helicopter had test instrumentation and otherfittings, the parasite drag of the helicopter is therefore rep-resented using a quadratic function available in (Ref.19):D/q( f t2) = 35.14+0.016(1.66αs)
2 where,q is the dynamicpressure andαs represents the helicopter pitch attitude.
Only linear aerodynamics is considered for Westland Lynxdue to unavailability of the actual airfoil table. Drees model isused for inflow calculation and blade flap dynamics equationsare solved using Newmark’s algorithm.
The relevant parameters used for helicopter dynamicsmodeling are shown in Table1.
Table 1. Helicopter parameters used for inverse simulationMain Rotor UH-60A Westland LynxNumber of Blades, N 4 4Radius,R (m) 8.17 6.4Blade Chord,c (m) 0.52 0.391Rotational Speed,Ω (rad/s) 27.01 35.63Longitudinal Shaft Tilt (degs.) -3 0Linear Blade Twist,θtw (rad/m) -0.3142 -0.14Lock Number,γ 6.33 7.12Blade Flap Frequency,νβ 1.04 1.07FuselageGross Weight (kg) 7876.18 4313.7Roll Inertia,Ixx (kg-m2) 4659 2767.1Pitch Inertia,Iyy (kg-m2) 38512 13904.5Yaw Inertia,Izz (kg-m2) 36800 12208.8Product of Inertia,Ixz (kg-m2) 1882 2034.8Tail RotorNumber of Blades,Nt 4 4Radius,Rt (m) 1.6764 1.106Blade Chord,ct (m) 0.24 0.180Rotational Speed,Ωt (rad/s) 124.62 193.103Linear Blade Twist,θtw (rad/m) 0 0
Maneuver Model
Since the purpose of inverse simulation is to derive the timehistory of controls required by the system model to respond asper the pre-defined output, the definition of desired output is avery important step. As explained in (Ref.20), the maneuvertrajectories can be typically modeled as smooth polynomialfunctions of time in terms of velocity of vehicle defined inearth fixed frame of reference. In the present analysis fivemaneuvers namely pull-up, pop-up, hurdle-hop, bob-up andpirouette maneuvers are simulated.
Pop-up Maneuver
As shown in Fig.2, the pop-up maneuver involves clear-ing an obstacle by a rapid controlled change of altitude oversome horizontal distance. The trajectory can be defined by apolynomial equation with some boundary conditions. A fifth-order polynomial has been found to satisfy these boundary
4
conditions (Ref.20) and is shown below.
Ze(t) = −h
[
6
(
ttm
)5
−15
(
ttm
)4
+10
(
ttm
)3]
(16)
Xe(t) =√
V 2f − Ze(t)2 (17)
h
x
Ze
Xe
Fig. 2. Picture depicting pop-up maneuver
A pop maneuver of 8 second duration is chosen for thepresent simulation, which is initiated from a steady velocityof 41.15m/s to reach an altitude of 25m. It is a constantforward velocity maneuver in which the resultant velocity oflogitudinal and vertical motions remain constant throughoutthe maneuver. The helicopter is accelerated over first half du-ration of maneuver which is then followed by a decelerationin vertical direction resulting in a gain of desired altitude.Thehelicopter would come back to its initial steady velocity stateby the end of maneuver. The variation of longitudinal and ver-tical non-dimensional acceleration of helicopter is shownas afunction of time in Fig.3
0 1 2 3 4 5 6 7 8 9−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Time (s)
Lo
ad F
acto
r (g
)
LongitudinalVertical
Fig. 3. Load factor variation for Pop-up maneuver
Hurdle-hop Maneuver
The hurdle-hop maneuver, as shown in Fig.4, involvesclearing an obstacle of heighth and returning back to theoriginal altitude over some specified distance,x. A polyno-mial equation describing the hurdle-hop maneuver is givenas (Ref.20):
Ze(t) =−64h
[
(
ttm
)3
−3
(
ttm
)2
+3
(
ttm
)
−1
]
(
ttm
)3
(18)
Xe(t) =√
V 2f − Ze(t)2 (19)
x
hZe
Xe
Fig. 4. Picture depicting hurdle-hop maneuver
For the present simulation this maneuver starts with asteady velocity of 41.15m/s and reaches an altitude of 50m during the first half of the maneuver. Like pop-up maneu-ver this is also a constant forward velocity maneuver in whichclimbing velocity becomes zero at the highest altitude point,from where helicopter starts descending and gradually reachesback to its initial altitude. This maneuver is carried out overa period of 17.5 seconds. The variation of longitudinal andvertical non-dimensional acceleration of helicopter is shownas a function of time in Fig.5
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (s)
Lo
ad f
acto
r (g
)
LongitudinalVertical
Fig. 5. Load factor variation for hurdle-hop maneuver
Pull-up Maneuver
Pull-up maneuver simulated here is for UH-60A BlackHawk helicopter. This is a terrain avoidance maneuver andis initiated from a high speed steady flight by pitching-up thehelicopter for a rapid gain in altitude. Unlike the pop-up ma-neuver, in which collective is initiated to accelerate vertically,longitudinal shaft tilt is initiated to pitch-up the helicopter that
5
Ze
Xe
h
Fig. 6. Picture depicting pull-up maneuver
results in an upward tilt in rotor thrust vector causes verticalacceleration. Pull-up maneuver is depicted in Fig.6
The maneuver being simulated is the counter 11029 flightfrom UH-60A flight test database. During this maneuver amaximum load factor of 2.12g was attained. The maneuverlasted for 9 seconds covering 40 rotor revolutions. The flighttest data is taken from (Refs.21,22). The measured load fac-tor and velocity ratio are shown in Figs.7 and8. One of thekey requirements of this maneuver was to maintain a load fac-tor of 1.75g for 3 seconds with less than 15m/s loss in air-speed. The helicopter attitude angles and angular rates areshown in Figs.9 and10 respectively, with negative represent-ing the nose down attitude.
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
Rotor revolution
Mea
n no
rmal
acc
eler
atio
n, g
3 secs. held at 1.75gwith less than 30 knotsloss in airspeed
Level Flight
Fig. 7. Picture depicting Measured mean load factor forpull-up maneuver
Like any other constant forward velocity maneuver, trajec-tory of pull-up maneuver also can be defined by a polynomialequation. The available information is used to derive the equa-tion for pull-up maneuver under simulation:
Ze(t) =−0.047(t)4+0.507(t)3+7.21(t)2
−2.429t+1.9725(20)
Xe(t) =√
V 2f − Ze(t)2 (21)
The variation of longitudinal and vertical non-dimensionalacceleration of helicopter, for pull-up maneuver, is shownasa function of time in Fig.11
0 10 20 30 40 50 60 700.2
0.25
0.3
0.35
0.4
Rotor Revolutions
Vel
ocity
rat
io, V
/ V tip
3 secs. held at 1.75gwith less than 30 knotsloss in airspeed
Fig. 8. Picture depicting velocity ratio for pull-up maneu-ver
0 10 20 30 40−10
−5
0
5
10
15
20
25
30
35
Rotor Revolutions
Ang
les,
deg
Roll
Pitch
Yaw
Fig. 9. Picture depicting helicopter attitude for pull-up ma-neuver
Bob-up Maneuver
As shown in Fig.12 helicopter starts from hover and ac-celerates vertically till half of the maneuver time then decel-erates to hover again by the end of the maneuver, when he-licopter reaches its desired altitude. Thus helicopter reachesmaximum velocity during the middle of the maneuver. Thevelocity of helicopter as a function of time, over this maneu-ver, is described by the polynomial equation (Ref.20):
V (t) =Vmax
[
−64
(
ttm
)6
+192
(
ttm
)5
−192
(
ttm
)4
+64
(
ttm
)3] (22)
Xe(t) = 0 (23)
6
0 10 20 30 40−10
−5
0
5
10
15
Rotor revolution
Ang
ular
rat
es, d
egs.
/sec
.
Pitch rate
Roll rate
Yaw rate
Fig. 10. Picture depicting helicopter attitude for pull-upmaneuver
0 5 10 15 20 25 30 35 40 45−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Revolution Number
Lo
ad F
acto
r (g
)
LongitudinalVertical
Fig. 11. Load factor variation for pull-up maneuver
Helicopter velocity profile for this maneuver is shown inFig. 13
Pirouette Maneuver
The pirouette maneuver starts from a hover condition ona reference circumference of a pre-defined radius; the rotor-craft must then perform a lateral translation around the cir-cumference, pointing its nose to the centre of the circle. Thismaneuver is depicted in Fig.14
This maneuver is described as
Xe(t) = −rcos(ωt) (24)
Ye(t) = rsin(ωt) (25)
Ze(t) = 0 (26)
ψ(t) = −ωt (27)
For the present simulation, a circle of radius(r) of 4 m withaircraft angular Velcoity(ω) of 0.5 rad/s, is chosen to carry
h
Xe
Ze
Fig. 12. Picture depicting bob-up maneuver
0 1 2 3 4 5 60
1
2
3
4
5
6
Vel
oci
ty (
m/s
)
Time (s)
Fig. 13. Velocity profile for bob-up maneuver
out this maneuver. The required acceleration profile to beachieved by helicopter is depicted in Fig.15
The effective parameters of various maneuvers under sim-ulation are as given in Table2
Table 2. Helicopter parameters used for inverse simulationManeuver V f (m/s) tm (s) h (m)Pull-up 77.76 9 152.4Pop-up 41.16 8 25Hurdle-hop 41.16 17.5 50Bob-up NA 5 NAPirouette NA 12.5 NA
RESULTS
First, the inverse flight dynamics simulation is carried outforthe pull-up maneuver for UH-60A helicopter as the flight testdata is available in the public domain. This allows for the val-idation of overall methodology. Also, the analysis is carriedout using linear as well as non-linear aerodynamics and theeffect of non-linear aerodynamic modelling is understood.
7
Fig. 14. Picture depicting pirouette maneuver
Pull-up Maneuver
The time history of measured and predicted collective, lat-eral cyclic and longitudinal cyclic angles are shown inFigs.16(a), 16(b)and16(c) respectively. The flight test col-lective angle shown in Fig.16(a)is observed to remain con-stant during initial steady flight regime (1–5 revolutions)ofthe maneuver. The predicted results show similar trend dur-ing this portion of the maneuver. During the rotor revolu-tion number 10–19, which represents a high velocity and highload factor regime of the maneuver, a significant reduction inrequired collective is observed despite of the increase in heli-copter load factor. This is probably due to the additional in-flow at the rotor due to high upward velocity of the helicopterand the analysis is able to predict this trend quite accurately.During revolutions 20–35 a gradual increase in collective con-trol is observed despite reduction in load factor owing to thefact that additional inflow that was available to the rotor isreduced due to decrease in vertical flight velocity. Since he-licopter tried to recover to its normal steady flight during lastfew revolutions (i.e., 32 to 40) of the maneuver, the requiredcollective increases at a gradual rate. The inverse simulationis unable to predict this gradual increase in collective anglebeyond revolution 28, instead the collective angle starts to de-crease again. This may be due to the use of simple inflowmodel which may not be able to accurately account for thechange in inflow experienced by the rotor during attitude re-covery phase of the maneuver. The predicted lateral cyclic an-gles shown in Fig.16(b)have similar trend as flight test, theprediction with linear aerodynamics shows best correlation interms of peak-to-peak magnitude and trend of variation dur-ing the maneuver. The predicted control angles using airfoillook-up table based non-linear aerodynamics show a decrease
0 2 4 6 8 10 12 14−1
−0.5
0
0.5
1
1.5
Revolution Number
Acc
eler
atio
n (
m/s
2 )
LongitudinalLateral
Fig. 15. Acceleration profile for pirouette maneuver
in lateral cyclic angle near revolution 9–15, and shows higherpeak-to-peak variation than flight test. The time history ofpredicted longitudinal cyclic angles using the two aerodynam-ics models is compared in Fig.16(c). Both analyses capturethe trend observed in the flight test data satisfactorily, but thepredictions show significant mean offset when compared toflight test. The magnitude of the longitudinal cyclic mean es-timated using linear aerodynamics is overpredicted by 2.9
and that using non-linear aerodynamics is overpredicted by3.7 approximately. The predictions with corrected mean val-ues are also shown in the Fig.16(c) for comparing the peak-to-peak magnitude variation. The prediction for longitudinalcyclic obtained using non-linear aerodynamics has peak-to-peak variation similar to that observed for flight test. In gen-eral, the predicted control angles obtained using non-linearstatic stall based aerodynamics shows larger variation in peak-to-peak magnitude. This may be due to decrease in lift withincrease in angle of attack with onset of static stall at vari-ous blade sections which may require the whole blade to beoperated at overall higher pitch angles.
Figure17 shows the longitudinal and lateral shaft tilt an-gles predicted by the inverse simulation. Both predictionsshow trends similar to flight test. The peak-to-peak magni-tudes are significantly underpredicted. The predictions fromthe linear aerodynamics show better correlation than the pre-dictions obtained using non-linear aerodynamics.
Fig. 18 compares the estimated helicopter body force tar-gets with the actual forces achieved during the inverse simu-lation. Corresponding roll and pitch moment comparison isshown in Fig.19. It is observed that the hub-forces and mo-ments achieved by the inverse simulation are nearly identicalto the estimated target loads from flight test data.
It appears that using a non-linear static stall based aerody-namic model for an unsteady pull-up maneuver characterizedby deep stall condition is not a good choice, and therefore,linear aerodynamic model without stall is better suited forun-steady maneuver analysis when compared to static airfoil ta-ble lookup. Incorporation of unsteady aerodynamics may be amore appropriate refinement. For all subsequent analysis for
8
0 10 20 30 400
5
10
15
Rotor Revolutions
Col
lect
ive,
θ0 (
deg)
Flight Test
Linear aero
Nonlinear aerodynamics
(a) Collective,θ0
0 10 20 30 40−5
0
5
10
Late
ral C
yclic
, θ1c
(de
g)
Flight TestLinear aerodynamicsNon−linear aerodynamics
(b) Lateral Cyclic,θ1c
0 10 20 30 40−15
−10
−5
0
Rotor Revolutions
Long
itudi
nal C
yclic
, θ1s
(de
g)
Flight TestLinear aerodynamicsNonlinear aerodynamics
With corrected steady value
Actual prediction
(c) Longitudinal Cyclic,θ1s
Fig. 16. Measured and predicted control angles using in-verse simulation for pull-up maneuver for UH-60A heli-copter
0 10 20 30 40−20
−15
−10
−5
0
5
10
15
20
25
Rotor Revolutions
Long
itudi
nal S
haft
Tilt
, αs (
deg)
Flight TestLinear aerodynamicsNonlinear aerodynamics
(a) Longitudinal shaft tilt,αs
0 10 20 30 40−15
−10
−5
0
5
10
15
20
Rotor Revolutions
Late
ral S
haft
Tilt
, φs (
deg)
Flight Test
Linear aerodynamics
Non−linear aerodynamics
(b) Lateral shaft tilt,αc
Fig. 17. Measured and predicted shaft tilt angles using in-verse simulation for pull-up maneuver
Lynx helicopter, linear aerodynamics model is used.
Since the derived sequence of controls are matching satis-factorily with the flight test results, other maneuvers, thosementioned in previous section, are simulated for WestlandLynx helicopter.
Pop-up Maneuver
Pop-up maneuver is initiated by increasing the collective tolift-up the helicopter in the initial phase of the maneuver inwhich helicopter is accelerated vertically as shown in Fig.20.It is also observed that, the lateral forces are of low magnitudeand hence the variation of lateral cyclic,θ1c is small. The in-creased collective makes the helicopter climb rapidly whichwould result in forward acceleration. The tendency of the tippath plane is to tilt backwards due to increase in lift on theadvancing side and decrease in lift on the retreating side. Thisis compensated by increasing the magnitude of negative lon-gitudinal cyclic as shown in Fig.20. The present analysis isable to capture this trend and shows fair correlation with thepredicted results from (Ref.23).
9
0 10 20 30 40−60000
−50000
−40000
−30000
−20000
−10000
0
10000lb
s.
Rotor Revolutions
Fx (Target)
Fy (Target)
Fz (Target)
Fx (Simulation)
Fy (Simulation)
Fz (Simulation)
Fig. 18. Calculated and achieved helicopter hub forces inbody fixed frame using measured flight test data
0 10 20 30 40−3000
−2000
−1000
0
1000
2000
lbs.
−ft
Rotor Revolutions
Pitch Moment (Target)Roll Moment (Target)Pitch Moment (simulation)Roll Moment (simulation)
Fig. 19. Calculated and achieved helicopter hub momentsin body fixed frame using measured flight test data
The simulated vehicle attitudes (roll and pitch) for this ma-neuver are shown in Fig.21. As mentioned earlier pop-upmaneuver is primarily a 2D maneuver where there is very lit-tle lateral motion as shown by nearly constant lateral attitudeduring this maneuver. The control variation discussed aboveresult in nose-up pitching motion in the initial phase (up to12 revolutions) of flight. A gradual reduction in vertical ac-celeration combined with reduction in forward decelerationmakes the helicopter return to its initial pitch attitude byhalftime point of maneuver when load factor on helicopter reachesback to its initial steady state value. The helicopter pitchori-entation during the second half of the maneuver is oppositeto that of the first half as the acceleration varies in oppositemanner. Helicopter returns to its initial orientation of steadystate by the end of the maneuver.
As there is no lateral motion in the simulated pop-up ma-neuver, tail rotor has to just counteract the main rotor torque
0 10 20 30 40 50−10
−5
0
5
10
15
Revolution Number
Co
ntr
ol A
ng
le(d
egre
e)
θ0
θ1c
θ1s
Reference simulation (Lu, 2007)
Present simulation
Fig. 20. Simulated and reference (Lu.,L, (Ref.23)) controlsfor pop-up maneuver
0 10 20 30 40 50−2
0
2
4
6
8
10
Revolution Number
Sh
aft
ang
le (
deg
ree)
α
s
φs
Fig. 21. Simulated shaft angles for pop-up maneuver
which is mainly a function of main rotor collective. There-fore, the variation of tail rotor collective shown in Fig.22follows the trend of the required main rotor collective. Thevariation of simulated tail rotor collective is similar to thatobtained by (Ref.23). The peak-to-peak magnitude is under-predicted by 1. This may be because of the differences inthe tail rotor modelling between the two analyses. The exactdetails of tail rotor model used in (Ref.23) is not available.
Hurdle-hop Maneuver
In hurdle-hop maneuver helicopter tries to regain its initial al-titude following a gain in altitude during the first half of themaneuver. Figure23shows the predicted time history of con-trol angles required to simulate this maneuver. It is observedthat the maneuver requires a gradual increase in vertical ac-celeration during first quarter (up to 24 rotor revolutions)ofmaneuver requiring an increase in collective pitch which isthen followed by a deceleration until it reaches the apex ofthe trajectory at the desired altitude. Hence collective isre-duced during the second quarter. The third quarter witnessesfurther reduction in collective followed by an increase, which
10
0 10 20 30 40 50
2.5
3
3.5
4
4.5
Revolution Number
Tai
l Ro
tor
Co
llect
ive
(deg
ree)
Referenece simulation (Lu, 2007)Present simulation
Fig. 22. Simulated tail rotor collective for pop-up maneu-ver
continues till the end of the maneuver as the helicopter triesto attain the original trim state. The longitudinal cyclic has tobe adjusted which requires initial decrease and then increasefollowed by further reduction to allow the helicopter to firstdecelerate during the climb process and then after the correctaltitude is attained it accelerates on its way back to original al-titude. The trends for collective and longitudinal cyclic anglesare similar to that predicted by (Ref.23). The lateral cyclicangle shows greater peak-to-peak variation than the referencepredictions. The reason for this is not fully understood.
0 20 40 60 80 100−10
−5
0
5
10
15
Revolution Number
Co
ntr
ol (
deg
ree)
θ1c
θ1s
θ0
Reference simulation (Lu, 2007)
Present simulation
Fig. 23. Simulated and reference (Lu.,L, (Ref.23)) controlsfor hurdle-hop maneuver
The simulated vehicle attitudes for this maneuver areshown in Fig.24. Helicopter is initially pitching-up to be con-sistent with the changing moments due to increasing verticalacceleration and corresponding forward deceleration which isthen followed by a gradual reduction in pitch angle to meet therequired forward acceleration and corresponding verticalde-celeration leading to initial helicopter orientation by the timehelicopter reaches the apex of the maneuver where longitudi-nal velocity would be maximum again with zero climb veloc-
0 20 40 60 80 100−4
−2
0
2
4
6
8
10
Revolution Number
Sh
aft
An
gle
(d
egre
e)
α
s
φs
Fig. 24. Simulated shaft angles for hurdle-hop maneuver
ity.
Hurdle-hop maneuver is also primarily a longitudinal ma-neuver, therefore the tail rotor collective follows the main ro-tor torque requirement which in turn follows the main rotorcollective as shown in Fig.25.
0 20 40 60 80 100−1
0
1
2
3
4
5
6
Revolution Number
Tai
l Ro
tor
Co
llect
ive
(deg
ree)
Reference simulation(Lu,2007)Present simulation
Fig. 25. Simulated shaft angles for hurdle-hop maneuver
Bob-up Maneuver
The bob-up maneuver is practically a 1D maneuver involv-ing axial flight, in which the helicopter is in hover conditionfrom which it is moved to hover condition at higher altitude.This requires the helicopter to accelerate vertically upwardsand then decelerate to return to hover. This is achieved bychanging the collective in near sinusoidal manner as shown inFig. 26. Thus the aircraft acceleration is zero around the mid-dle of the maneuver after which helicopter starts deceleratingrequiring a decrease in collective. Variation in cyclic controlsis negligible due to the absence of any longitudinal or lateralmotion during the maneuver.
The variation of longitudinal and lateral shaft tilt anglesfor Bob-up maneuver is shown in Fig.27. As expected the at-
11
0 5 10 15 20 25 30−2
0
2
4
6
8
10
12
14
Revolution Number
Co
ntr
ol (
deg
ree)
θ
0
θ1c
θ1s
Fig. 26. Simulated controls for bob-up maneuver
0 5 10 15 20 25 30−4
−3
−2
−1
0
1
2
3
4
5
Revolution Number
Sh
aft
An
gle
(d
egre
e)
αs
φs
Fig. 27. Simulated shaft angles for bob-up Maneuver
titude of the helicopter remains unchanged during the maneu-ver. The variation of tail rotor collective is shown in Fig.28.During the initial phase of maneuver collective is gradually in-creased to balance the increasing main rotor torque. After theinitial acceleration phase tail rotor collective also decreasesalong with decreasing power requirement from main rotor.
Pirouette Maneuver
Since the maneuver is of very low speed with a resultant ve-locity of 2 m/s in horizontal plane with a rotational speed of0.5 rad/s, the collective is nearly constant through out thema-neuver. The periodic velocity in longitudinal and lateral direc-tions result in periodically varying cycling controls as shownin Fig. 29.
The simulated shaft angles for pirouette maneuver shownin Fig. 30variation of longitudinal shaft tilt reflects the varia-tion of longitudinal acceleration required by maneuver. Sincecollective remains nearly constant, tip path plane has to betilted forward and side-wards to accelerate the vehicle dur-ing first half of the maneuver and reversed back to its initialorientation by end of the maneuver. Similarly lateral shafttilt varies periodically to achieve periodically varying lateral
0 5 10 15 20 25 301
1.5
2
2.5
3
3.5
4
Revolution Number
Tai
l Ro
tor
Co
llect
ive
(deg
ree)
Fig. 28. Simulated tail rotor collective for bob-up maneu-ver
force to enable the flight in a circular path. The variation oftail rotor collective is shown in Fig.31. The tail rotor col-lective plays a crucial role in maintaining a constant yaw rateduring this maneuver.
0 20 40 60 80−4
−2
0
2
4
6
8
10
12
Revolution Number
Con
trol
Ang
les,
deg
θ0
θ1c
θ1s
Fig. 29. Simulated controls for pirouette maneuver
SUMMARY & CONCLUSIONS
The implementation of integration based inverse simulationalgorithm is discussed in this paper. The analysis is validatedby simulating UH-60A Black Hawk helicopter for 2.1g dy-namic pull-up maneuver. The results for various maneuvers,including pop-up and hurdle-hop maneuvers, are simulatedfor controls time history for Westland Lynx helicopters andcompared with simulation results available in literature.Fi-nally, bob-up and pirouette maneuver is also analyzed. Theconclusions drawn from this analysis are listed below:
1. A simple helicopter dynamics model consisting of artic-ulated blades with linear twist, quasi-steady linear aero-dynamics with Drees inflow model is adequate to study
12
0 20 40 60 80−10
−5
0
5
10
15
Rotor Revolutions
Sha
ft A
ngle
s, d
eg
αs
φs
Fig. 30. Simulated shaft angles for pirouette maneuver
0 10 20 30 40 50 60 70 801.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
Revolution Number
Tai
l Ro
tor
Co
llect
ive
(deg
ree)
Fig. 31. Simulated tail rotor collective for pirouette ma-neuver
the behaviour of helicopter performing unsteady maneu-vers.
2. The generalized intergation based inverse simulation issuccessfully validated and verified for unsteady maneu-vers performed by two different helicopters and five dif-ferent maneuver. The predicted controls show fair cor-relation with flight test controls (in the case of pull-upmaneuver) and reference controls in magnitude and vari-ation. However, magnitudes of tail rotor collective areunderpredicted, possibly due to differences in tail rotormodelling.
3. For all symmetric maneuvers such as pull-up, pop-up,bob-up, and hurdle-hop maneuvers the tail rotor collec-tive follows the trend of collective pitch control.
4. It is observed that the variation of pitch orientation fol-lows the trend of the vertical acceleration of the heli-copter.
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